Bugeaud, Yann, Mignotte, Maurice and Siksek, Samir (2006) Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers. Annals of Mathematics, Vol.163 (No.3). pp. 969-1018. doi:10.4007/annals.2006.163.969

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Abstract

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Annals of Mathematics
Publisher:	Mathematical Sciences Publishers
ISSN:	0003-486X
Official Date:	May 2006
Dates:	Date Event May 2006 Published
Volume:	Vol.163
Number:	No.3
Number of Pages:	50
Page Range:	pp. 969-1018
DOI:	10.4007/annals.2006.163.969
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Sultan Qaboos University (Oman)
Grant number:	IG/SCI/DOMS/02/06

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